\(\int \frac {a+b x+c x^2}{(d+e x)^3 (f+g x)^{3/2}} \, dx\) [832]

Optimal result

Integrand size = 27, antiderivative size = 248 \[ \int \frac {a+b x+c x^2}{(d+e x)^3 (f+g x)^{3/2}} \, dx=\frac {2 \left (c f^2-b f g+a g^2\right )}{(e f-d g)^3 \sqrt {f+g x}}-\frac {\left (c d^2-b d e+a e^2\right ) \sqrt {f+g x}}{2 e (e f-d g)^2 (d+e x)^2}+\frac {(c d (8 e f-d g)-e (4 b e f+3 b d g-7 a e g)) \sqrt {f+g x}}{4 e (e f-d g)^3 (d+e x)}-\frac {\left (c \left (8 e^2 f^2+8 d e f g-d^2 g^2\right )+3 e g (5 a e g-b (4 e f+d g))\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{4 e^{3/2} (e f-d g)^{7/2}} \]

[Out]

-1/4*(c*(-d^2*g^2+8*d*e*f*g+8*e^2*f^2)+3*e*g*(5*a*e*g-b*(d*g+4*e*f)))*arctanh(e^(1/2)*(g*x+f)^(1/2)/(-d*g+e*f)
^(1/2))/e^(3/2)/(-d*g+e*f)^(7/2)+2*(a*g^2-b*f*g+c*f^2)/(-d*g+e*f)^3/(g*x+f)^(1/2)-1/2*(a*e^2-b*d*e+c*d^2)*(g*x
+f)^(1/2)/e/(-d*g+e*f)^2/(e*x+d)^2+1/4*(c*d*(-d*g+8*e*f)-e*(-7*a*e*g+3*b*d*g+4*b*e*f))*(g*x+f)^(1/2)/e/(-d*g+e
*f)^3/(e*x+d)

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {911, 1273, 467, 464, 214} \[ \int \frac {a+b x+c x^2}{(d+e x)^3 (f+g x)^{3/2}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (3 e g (5 a e g-b (d g+4 e f))+c \left (-d^2 g^2+8 d e f g+8 e^2 f^2\right )\right )}{4 e^{3/2} (e f-d g)^{7/2}}-\frac {\sqrt {f+g x} \left (a e^2-b d e+c d^2\right )}{2 e (d+e x)^2 (e f-d g)^2}+\frac {2 \left (a g^2-b f g+c f^2\right )}{\sqrt {f+g x} (e f-d g)^3}+\frac {\sqrt {f+g x} (c d (8 e f-d g)-e (-7 a e g+3 b d g+4 b e f))}{4 e (d+e x) (e f-d g)^3} \]

[In]

Int[(a + b*x + c*x^2)/((d + e*x)^3*(f + g*x)^(3/2)),x]

[Out]

(2*(c*f^2 - b*f*g + a*g^2))/((e*f - d*g)^3*Sqrt[f + g*x]) - ((c*d^2 - b*d*e + a*e^2)*Sqrt[f + g*x])/(2*e*(e*f
- d*g)^2*(d + e*x)^2) + ((c*d*(8*e*f - d*g) - e*(4*b*e*f + 3*b*d*g - 7*a*e*g))*Sqrt[f + g*x])/(4*e*(e*f - d*g)
^3*(d + e*x)) - ((c*(8*e^2*f^2 + 8*d*e*f*g - d^2*g^2) + 3*e*g*(5*a*e*g - b*(4*e*f + d*g)))*ArcTanh[(Sqrt[e]*Sq
rt[f + g*x])/Sqrt[e*f - d*g]])/(4*e^(3/2)*(e*f - d*g)^(7/2))

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 464

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[c*(e*x)^(m +
1)*((a + b*x^n)^(p + 1)/(a*e*(m + 1))), x] + Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)), In
t[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[n] ||
GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) &&  !ILtQ[p, -1]

Rule 467

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[(-a)^(m/2 - 1)*(b*c - a*d)*x
*((a + b*x^2)^(p + 1)/(2*b^(m/2 + 1)*(p + 1))), x] + Dist[1/(2*b^(m/2 + 1)*(p + 1)), Int[x^m*(a + b*x^2)^(p +
1)*ExpandToSum[2*b*(p + 1)*Together[(b^(m/2)*(c + d*x^2) - (-a)^(m/2 - 1)*(b*c - a*d)*x^(-m + 2))/(a + b*x^2)]
- ((-a)^(m/2 - 1)*(b*c - a*d))/x^m, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &
& ILtQ[m/2, 0] && (IntegerQ[p] || EqQ[m + 2*p + 1, 0])

Rule 911

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> With[{q = Denominator[m]}, Dist[q/e, Subst[Int[x^(q*(m + 1) - 1)*((e*f - d*g)/e + g*(x^q/e))^n*((c*d^2 - b*d
*e + a*e^2)/e^2 - (2*c*d - b*e)*(x^q/e^2) + c*(x^(2*q)/e^2))^p, x], x, (d + e*x)^(1/q)], x]] /; FreeQ[{a, b, c
, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegersQ[n,
p] && FractionQ[m]

Rule 1273

Int[(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp[(-d)^(m
/2 - 1)*(c*d^2 - b*d*e + a*e^2)^p*x*((d + e*x^2)^(q + 1)/(2*e^(2*p + m/2)*(q + 1))), x] + Dist[(-d)^(m/2 - 1)/
(2*e^(2*p)*(q + 1)), Int[x^m*(d + e*x^2)^(q + 1)*ExpandToSum[Together[(1/(d + e*x^2))*(2*(-d)^(-m/2 + 1)*e^(2*
p)*(q + 1)*(a + b*x^2 + c*x^4)^p - ((c*d^2 - b*d*e + a*e^2)^p/(e^(m/2)*x^m))*(d + e*(2*q + 3)*x^2))], x], x],
x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && ILtQ[q, -1] && ILtQ[m/2, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {2 \text {Subst}\left (\int \frac {\frac {c f^2-b f g+a g^2}{g^2}-\frac {(2 c f-b g) x^2}{g^2}+\frac {c x^4}{g^2}}{x^2 \left (\frac {-e f+d g}{g}+\frac {e x^2}{g}\right )^3} \, dx,x,\sqrt {f+g x}\right )}{g} \\ & = -\frac {\left (c d^2-b d e+a e^2\right ) \sqrt {f+g x}}{2 e (e f-d g)^2 (d+e x)^2}-\frac {g^3 \text {Subst}\left (\int \frac {\frac {4 e^2 (e f-d g) \left (c f^2-b f g+a g^2\right )}{g^5}-\frac {e \left (3 e (b d-a e) g^2+c \left (4 e^2 f^2-8 d e f g+d^2 g^2\right )\right ) x^2}{g^5}}{x^2 \left (\frac {-e f+d g}{g}+\frac {e x^2}{g}\right )^2} \, dx,x,\sqrt {f+g x}\right )}{2 e^2 (e f-d g)^2} \\ & = -\frac {\left (c d^2-b d e+a e^2\right ) \sqrt {f+g x}}{2 e (e f-d g)^2 (d+e x)^2}+\frac {(c d (8 e f-d g)-e (4 b e f+3 b d g-7 a e g)) \sqrt {f+g x}}{4 e (e f-d g)^3 (d+e x)}+\frac {g^3 \text {Subst}\left (\int \frac {\frac {8 e^2 \left (c f^2-b f g+a g^2\right )}{g^4}+\frac {e (c d (8 e f-d g)-e (4 b e f+3 b d g-7 a e g)) x^2}{g^3 (e f-d g)}}{x^2 \left (\frac {-e f+d g}{g}+\frac {e x^2}{g}\right )} \, dx,x,\sqrt {f+g x}\right )}{4 e^2 (e f-d g)^2} \\ & = \frac {2 \left (c f^2-b f g+a g^2\right )}{(e f-d g)^3 \sqrt {f+g x}}-\frac {\left (c d^2-b d e+a e^2\right ) \sqrt {f+g x}}{2 e (e f-d g)^2 (d+e x)^2}+\frac {(c d (8 e f-d g)-e (4 b e f+3 b d g-7 a e g)) \sqrt {f+g x}}{4 e (e f-d g)^3 (d+e x)}+\frac {\left (c \left (8 e^2 f^2+8 d e f g-d^2 g^2\right )+3 e g (5 a e g-b (4 e f+d g))\right ) \text {Subst}\left (\int \frac {1}{\frac {-e f+d g}{g}+\frac {e x^2}{g}} \, dx,x,\sqrt {f+g x}\right )}{4 e g (e f-d g)^3} \\ & = \frac {2 \left (c f^2-b f g+a g^2\right )}{(e f-d g)^3 \sqrt {f+g x}}-\frac {\left (c d^2-b d e+a e^2\right ) \sqrt {f+g x}}{2 e (e f-d g)^2 (d+e x)^2}+\frac {(c d (8 e f-d g)-e (4 b e f+3 b d g-7 a e g)) \sqrt {f+g x}}{4 e (e f-d g)^3 (d+e x)}-\frac {\left (c \left (8 e^2 f^2+8 d e f g-d^2 g^2\right )+3 e g (5 a e g-b (4 e f+d g))\right ) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{4 e^{3/2} (e f-d g)^{7/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.17 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.20 \[ \int \frac {a+b x+c x^2}{(d+e x)^3 (f+g x)^{3/2}} \, dx=\frac {\frac {\sqrt {e} \left (c \left (8 e^3 f^2 x^2+d^3 g (f+g x)+8 d e^2 f x (3 f+g x)+d^2 e \left (14 f^2+5 f g x-g^2 x^2\right )\right )-e \left (a \left (-8 d^2 g^2-d e g (9 f+25 g x)+e^2 \left (2 f^2-5 f g x-15 g^2 x^2\right )\right )+b \left (4 e^2 f x (f+3 g x)+d^2 g (13 f+5 g x)+d e \left (2 f^2+21 f g x+3 g^2 x^2\right )\right )\right )\right )}{(e f-d g)^3 (d+e x)^2 \sqrt {f+g x}}-\frac {\left (c \left (8 e^2 f^2+8 d e f g-d^2 g^2\right )+3 e g (5 a e g-b (4 e f+d g))\right ) \arctan \left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {-e f+d g}}\right )}{(-e f+d g)^{7/2}}}{4 e^{3/2}} \]

[In]

Integrate[(a + b*x + c*x^2)/((d + e*x)^3*(f + g*x)^(3/2)),x]

[Out]

((Sqrt[e]*(c*(8*e^3*f^2*x^2 + d^3*g*(f + g*x) + 8*d*e^2*f*x*(3*f + g*x) + d^2*e*(14*f^2 + 5*f*g*x - g^2*x^2))
- e*(a*(-8*d^2*g^2 - d*e*g*(9*f + 25*g*x) + e^2*(2*f^2 - 5*f*g*x - 15*g^2*x^2)) + b*(4*e^2*f*x*(f + 3*g*x) + d
^2*g*(13*f + 5*g*x) + d*e*(2*f^2 + 21*f*g*x + 3*g^2*x^2)))))/((e*f - d*g)^3*(d + e*x)^2*Sqrt[f + g*x]) - ((c*(
8*e^2*f^2 + 8*d*e*f*g - d^2*g^2) + 3*e*g*(5*a*e*g - b*(4*e*f + d*g)))*ArcTan[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[-(e*
f) + d*g]])/(-(e*f) + d*g)^(7/2))/(4*e^(3/2))

Maple [A] (verified)

Time = 0.65 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.12

method	result	size
pseudoelliptic	\(-\frac {15 \left (\sqrt {g x +f}\, \left (e x +d \right )^{2} \left (\left (a \,g^{2}-\frac {4}{5} b f g +\frac {8}{15} c \,f^{2}\right ) e^{2}-\frac {d g \left (b g -\frac {8 c f}{3}\right ) e}{5}-\frac {c \,d^{2} g^{2}}{15}\right ) \arctan \left (\frac {e \sqrt {g x +f}}{\sqrt {\left (d g -e f \right ) e}}\right )+\frac {8 \left (\left (\frac {15 a \,g^{2} x^{2}}{8}+\frac {5 \left (-\frac {12 b x}{5}+a \right ) x f g}{8}-\frac {f^{2} \left (-4 c \,x^{2}+2 b x +a \right )}{4}\right ) e^{3}+\frac {9 d \left (\left (-\frac {1}{3} b \,x^{2}+\frac {25}{9} a x \right ) g^{2}+f \left (\frac {8}{9} c \,x^{2}-\frac {7}{3} b x +a \right ) g -\frac {2 f^{2} \left (-12 c x +b \right )}{9}\right ) e^{2}}{8}+d^{2} \left (\left (a -\frac {1}{8} c \,x^{2}-\frac {5}{8} b x \right ) g^{2}-\frac {13 \left (-\frac {5 c x}{13}+b \right ) f g}{8}+\frac {7 c \,f^{2}}{4}\right ) e +\frac {c \,d^{3} g \left (g x +f \right )}{8}\right ) \sqrt {\left (d g -e f \right ) e}}{15}\right )}{4 \sqrt {g x +f}\, \sqrt {\left (d g -e f \right ) e}\, \left (d g -e f \right )^{3} \left (e x +d \right )^{2} e}\)	\(279\)
derivativedivides	\(-\frac {2 \left (a \,g^{2}-b f g +c \,f^{2}\right )}{\left (d g -e f \right )^{3} \sqrt {g x +f}}-\frac {2 \left (\frac {\left (\frac {7}{8} a \,e^{2} g^{2}-\frac {3}{8} b d e \,g^{2}-\frac {1}{2} b \,e^{2} f g -\frac {1}{8} c \,d^{2} g^{2}+c d e f g \right ) \left (g x +f \right )^{\frac {3}{2}}+\frac {g \left (9 a d \,e^{2} g^{2}-9 a \,e^{3} f g -5 b \,d^{2} e \,g^{2}+b d \,e^{2} f g +4 b \,e^{3} f^{2}+c \,d^{3} g^{2}+7 c \,d^{2} e f g -8 c d \,e^{2} f^{2}\right ) \sqrt {g x +f}}{8 e}}{\left (e \left (g x +f \right )+d g -e f \right )^{2}}+\frac {\left (15 a \,e^{2} g^{2}-3 b d e \,g^{2}-12 b \,e^{2} f g -c \,d^{2} g^{2}+8 c d e f g +8 c \,e^{2} f^{2}\right ) \arctan \left (\frac {e \sqrt {g x +f}}{\sqrt {\left (d g -e f \right ) e}}\right )}{8 e \sqrt {\left (d g -e f \right ) e}}\right )}{\left (d g -e f \right )^{3}}\)	\(294\)
default	\(-\frac {2 \left (a \,g^{2}-b f g +c \,f^{2}\right )}{\left (d g -e f \right )^{3} \sqrt {g x +f}}-\frac {2 \left (\frac {\left (\frac {7}{8} a \,e^{2} g^{2}-\frac {3}{8} b d e \,g^{2}-\frac {1}{2} b \,e^{2} f g -\frac {1}{8} c \,d^{2} g^{2}+c d e f g \right ) \left (g x +f \right )^{\frac {3}{2}}+\frac {g \left (9 a d \,e^{2} g^{2}-9 a \,e^{3} f g -5 b \,d^{2} e \,g^{2}+b d \,e^{2} f g +4 b \,e^{3} f^{2}+c \,d^{3} g^{2}+7 c \,d^{2} e f g -8 c d \,e^{2} f^{2}\right ) \sqrt {g x +f}}{8 e}}{\left (e \left (g x +f \right )+d g -e f \right )^{2}}+\frac {\left (15 a \,e^{2} g^{2}-3 b d e \,g^{2}-12 b \,e^{2} f g -c \,d^{2} g^{2}+8 c d e f g +8 c \,e^{2} f^{2}\right ) \arctan \left (\frac {e \sqrt {g x +f}}{\sqrt {\left (d g -e f \right ) e}}\right )}{8 e \sqrt {\left (d g -e f \right ) e}}\right )}{\left (d g -e f \right )^{3}}\)	\(294\)

[In]

int((c*x^2+b*x+a)/(e*x+d)^3/(g*x+f)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-15/4/(g*x+f)^(1/2)/((d*g-e*f)*e)^(1/2)*((g*x+f)^(1/2)*(e*x+d)^2*((a*g^2-4/5*b*f*g+8/15*c*f^2)*e^2-1/5*d*g*(b*
g-8/3*c*f)*e-1/15*c*d^2*g^2)*arctan(e*(g*x+f)^(1/2)/((d*g-e*f)*e)^(1/2))+8/15*((15/8*a*g^2*x^2+5/8*(-12/5*b*x+
a)*x*f*g-1/4*f^2*(-4*c*x^2+2*b*x+a))*e^3+9/8*d*((-1/3*b*x^2+25/9*a*x)*g^2+f*(8/9*c*x^2-7/3*b*x+a)*g-2/9*f^2*(-
12*c*x+b))*e^2+d^2*((a-1/8*c*x^2-5/8*b*x)*g^2-13/8*(-5/13*c*x+b)*f*g+7/4*c*f^2)*e+1/8*c*d^3*g*(g*x+f))*((d*g-e
*f)*e)^(1/2))/(d*g-e*f)^3/(e*x+d)^2/e

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 935 vs. \(2 (226) = 452\).

Time = 0.59 (sec) , antiderivative size = 1883, normalized size of antiderivative = 7.59 \[ \int \frac {a+b x+c x^2}{(d+e x)^3 (f+g x)^{3/2}} \, dx=\text {Too large to display} \]

[In]

integrate((c*x^2+b*x+a)/(e*x+d)^3/(g*x+f)^(3/2),x, algorithm="fricas")

[Out]

[-1/8*((8*c*d^2*e^2*f^3 + 4*(2*c*d^3*e - 3*b*d^2*e^2)*f^2*g - (c*d^4 + 3*b*d^3*e - 15*a*d^2*e^2)*f*g^2 + (8*c*
e^4*f^2*g + 4*(2*c*d*e^3 - 3*b*e^4)*f*g^2 - (c*d^2*e^2 + 3*b*d*e^3 - 15*a*e^4)*g^3)*x^3 + (8*c*e^4*f^3 + 12*(2
*c*d*e^3 - b*e^4)*f^2*g + 3*(5*c*d^2*e^2 - 9*b*d*e^3 + 5*a*e^4)*f*g^2 - 2*(c*d^3*e + 3*b*d^2*e^2 - 15*a*d*e^3)
*g^3)*x^2 + (16*c*d*e^3*f^3 + 24*(c*d^2*e^2 - b*d*e^3)*f^2*g + 6*(c*d^3*e - 3*b*d^2*e^2 + 5*a*d*e^3)*f*g^2 - (
c*d^4 + 3*b*d^3*e - 15*a*d^2*e^2)*g^3)*x)*sqrt(e^2*f - d*e*g)*log((e*g*x + 2*e*f - d*g + 2*sqrt(e^2*f - d*e*g)
*sqrt(g*x + f))/(e*x + d)) + 2*(8*a*d^3*e^2*g^3 - 2*(7*c*d^2*e^3 - b*d*e^4 - a*e^5)*f^3 + (13*c*d^3*e^2 + 11*b
*d^2*e^3 - 11*a*d*e^4)*f^2*g + (c*d^4*e - 13*b*d^3*e^2 + a*d^2*e^3)*f*g^2 - (8*c*e^5*f^3 - 12*b*e^5*f^2*g - 3*
(3*c*d^2*e^3 - 3*b*d*e^4 - 5*a*e^5)*f*g^2 + (c*d^3*e^2 + 3*b*d^2*e^3 - 15*a*d*e^4)*g^3)*x^2 - (4*(6*c*d*e^4 -
b*e^5)*f^3 - (19*c*d^2*e^3 + 17*b*d*e^4 - 5*a*e^5)*f^2*g - 4*(c*d^3*e^2 - 4*b*d^2*e^3 - 5*a*d*e^4)*f*g^2 - (c*
d^4*e - 5*b*d^3*e^2 + 25*a*d^2*e^3)*g^3)*x)*sqrt(g*x + f))/(d^2*e^6*f^5 - 4*d^3*e^5*f^4*g + 6*d^4*e^4*f^3*g^2
- 4*d^5*e^3*f^2*g^3 + d^6*e^2*f*g^4 + (e^8*f^4*g - 4*d*e^7*f^3*g^2 + 6*d^2*e^6*f^2*g^3 - 4*d^3*e^5*f*g^4 + d^4
*e^4*g^5)*x^3 + (e^8*f^5 - 2*d*e^7*f^4*g - 2*d^2*e^6*f^3*g^2 + 8*d^3*e^5*f^2*g^3 - 7*d^4*e^4*f*g^4 + 2*d^5*e^3
*g^5)*x^2 + (2*d*e^7*f^5 - 7*d^2*e^6*f^4*g + 8*d^3*e^5*f^3*g^2 - 2*d^4*e^4*f^2*g^3 - 2*d^5*e^3*f*g^4 + d^6*e^2
*g^5)*x), 1/4*((8*c*d^2*e^2*f^3 + 4*(2*c*d^3*e - 3*b*d^2*e^2)*f^2*g - (c*d^4 + 3*b*d^3*e - 15*a*d^2*e^2)*f*g^2
+ (8*c*e^4*f^2*g + 4*(2*c*d*e^3 - 3*b*e^4)*f*g^2 - (c*d^2*e^2 + 3*b*d*e^3 - 15*a*e^4)*g^3)*x^3 + (8*c*e^4*f^3
+ 12*(2*c*d*e^3 - b*e^4)*f^2*g + 3*(5*c*d^2*e^2 - 9*b*d*e^3 + 5*a*e^4)*f*g^2 - 2*(c*d^3*e + 3*b*d^2*e^2 - 15*
a*d*e^3)*g^3)*x^2 + (16*c*d*e^3*f^3 + 24*(c*d^2*e^2 - b*d*e^3)*f^2*g + 6*(c*d^3*e - 3*b*d^2*e^2 + 5*a*d*e^3)*f
*g^2 - (c*d^4 + 3*b*d^3*e - 15*a*d^2*e^2)*g^3)*x)*sqrt(-e^2*f + d*e*g)*arctan(sqrt(-e^2*f + d*e*g)*sqrt(g*x +
f)/(e*g*x + e*f)) - (8*a*d^3*e^2*g^3 - 2*(7*c*d^2*e^3 - b*d*e^4 - a*e^5)*f^3 + (13*c*d^3*e^2 + 11*b*d^2*e^3 -
11*a*d*e^4)*f^2*g + (c*d^4*e - 13*b*d^3*e^2 + a*d^2*e^3)*f*g^2 - (8*c*e^5*f^3 - 12*b*e^5*f^2*g - 3*(3*c*d^2*e^
3 - 3*b*d*e^4 - 5*a*e^5)*f*g^2 + (c*d^3*e^2 + 3*b*d^2*e^3 - 15*a*d*e^4)*g^3)*x^2 - (4*(6*c*d*e^4 - b*e^5)*f^3
- (19*c*d^2*e^3 + 17*b*d*e^4 - 5*a*e^5)*f^2*g - 4*(c*d^3*e^2 - 4*b*d^2*e^3 - 5*a*d*e^4)*f*g^2 - (c*d^4*e - 5*b
*d^3*e^2 + 25*a*d^2*e^3)*g^3)*x)*sqrt(g*x + f))/(d^2*e^6*f^5 - 4*d^3*e^5*f^4*g + 6*d^4*e^4*f^3*g^2 - 4*d^5*e^3
*f^2*g^3 + d^6*e^2*f*g^4 + (e^8*f^4*g - 4*d*e^7*f^3*g^2 + 6*d^2*e^6*f^2*g^3 - 4*d^3*e^5*f*g^4 + d^4*e^4*g^5)*x
^3 + (e^8*f^5 - 2*d*e^7*f^4*g - 2*d^2*e^6*f^3*g^2 + 8*d^3*e^5*f^2*g^3 - 7*d^4*e^4*f*g^4 + 2*d^5*e^3*g^5)*x^2 +
(2*d*e^7*f^5 - 7*d^2*e^6*f^4*g + 8*d^3*e^5*f^3*g^2 - 2*d^4*e^4*f^2*g^3 - 2*d^5*e^3*f*g^4 + d^6*e^2*g^5)*x)]

Sympy [F(-1)]

Timed out. \[ \int \frac {a+b x+c x^2}{(d+e x)^3 (f+g x)^{3/2}} \, dx=\text {Timed out} \]

[In]

integrate((c*x**2+b*x+a)/(e*x+d)**3/(g*x+f)**(3/2),x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {a+b x+c x^2}{(d+e x)^3 (f+g x)^{3/2}} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((c*x^2+b*x+a)/(e*x+d)^3/(g*x+f)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e*(d*g-e*f)>0)', see `assume?`
for more de

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 472 vs. \(2 (226) = 452\).

Time = 0.30 (sec) , antiderivative size = 472, normalized size of antiderivative = 1.90 \[ \int \frac {a+b x+c x^2}{(d+e x)^3 (f+g x)^{3/2}} \, dx=\frac {{\left (8 \, c e^{2} f^{2} + 8 \, c d e f g - 12 \, b e^{2} f g - c d^{2} g^{2} - 3 \, b d e g^{2} + 15 \, a e^{2} g^{2}\right )} \arctan \left (\frac {\sqrt {g x + f} e}{\sqrt {-e^{2} f + d e g}}\right )}{4 \, {\left (e^{4} f^{3} - 3 \, d e^{3} f^{2} g + 3 \, d^{2} e^{2} f g^{2} - d^{3} e g^{3}\right )} \sqrt {-e^{2} f + d e g}} + \frac {2 \, {\left (c f^{2} - b f g + a g^{2}\right )}}{{\left (e^{3} f^{3} - 3 \, d e^{2} f^{2} g + 3 \, d^{2} e f g^{2} - d^{3} g^{3}\right )} \sqrt {g x + f}} + \frac {8 \, {\left (g x + f\right )}^{\frac {3}{2}} c d e^{2} f g - 4 \, {\left (g x + f\right )}^{\frac {3}{2}} b e^{3} f g - 8 \, \sqrt {g x + f} c d e^{2} f^{2} g + 4 \, \sqrt {g x + f} b e^{3} f^{2} g - {\left (g x + f\right )}^{\frac {3}{2}} c d^{2} e g^{2} - 3 \, {\left (g x + f\right )}^{\frac {3}{2}} b d e^{2} g^{2} + 7 \, {\left (g x + f\right )}^{\frac {3}{2}} a e^{3} g^{2} + 7 \, \sqrt {g x + f} c d^{2} e f g^{2} + \sqrt {g x + f} b d e^{2} f g^{2} - 9 \, \sqrt {g x + f} a e^{3} f g^{2} + \sqrt {g x + f} c d^{3} g^{3} - 5 \, \sqrt {g x + f} b d^{2} e g^{3} + 9 \, \sqrt {g x + f} a d e^{2} g^{3}}{4 \, {\left (e^{4} f^{3} - 3 \, d e^{3} f^{2} g + 3 \, d^{2} e^{2} f g^{2} - d^{3} e g^{3}\right )} {\left ({\left (g x + f\right )} e - e f + d g\right )}^{2}} \]

[In]

integrate((c*x^2+b*x+a)/(e*x+d)^3/(g*x+f)^(3/2),x, algorithm="giac")

[Out]

1/4*(8*c*e^2*f^2 + 8*c*d*e*f*g - 12*b*e^2*f*g - c*d^2*g^2 - 3*b*d*e*g^2 + 15*a*e^2*g^2)*arctan(sqrt(g*x + f)*e
/sqrt(-e^2*f + d*e*g))/((e^4*f^3 - 3*d*e^3*f^2*g + 3*d^2*e^2*f*g^2 - d^3*e*g^3)*sqrt(-e^2*f + d*e*g)) + 2*(c*f
^2 - b*f*g + a*g^2)/((e^3*f^3 - 3*d*e^2*f^2*g + 3*d^2*e*f*g^2 - d^3*g^3)*sqrt(g*x + f)) + 1/4*(8*(g*x + f)^(3/
2)*c*d*e^2*f*g - 4*(g*x + f)^(3/2)*b*e^3*f*g - 8*sqrt(g*x + f)*c*d*e^2*f^2*g + 4*sqrt(g*x + f)*b*e^3*f^2*g - (
g*x + f)^(3/2)*c*d^2*e*g^2 - 3*(g*x + f)^(3/2)*b*d*e^2*g^2 + 7*(g*x + f)^(3/2)*a*e^3*g^2 + 7*sqrt(g*x + f)*c*d
^2*e*f*g^2 + sqrt(g*x + f)*b*d*e^2*f*g^2 - 9*sqrt(g*x + f)*a*e^3*f*g^2 + sqrt(g*x + f)*c*d^3*g^3 - 5*sqrt(g*x
+ f)*b*d^2*e*g^3 + 9*sqrt(g*x + f)*a*d*e^2*g^3)/((e^4*f^3 - 3*d*e^3*f^2*g + 3*d^2*e^2*f*g^2 - d^3*e*g^3)*((g*x
+ f)*e - e*f + d*g)^2)

Mupad [B] (verification not implemented)

Time = 12.17 (sec) , antiderivative size = 363, normalized size of antiderivative = 1.46 \[ \int \frac {a+b x+c x^2}{(d+e x)^3 (f+g x)^{3/2}} \, dx=\frac {\mathrm {atan}\left (\frac {\sqrt {f+g\,x}\,\left (-d^3\,e\,g^3+3\,d^2\,e^2\,f\,g^2-3\,d\,e^3\,f^2\,g+e^4\,f^3\right )}{\sqrt {e}\,{\left (d\,g-e\,f\right )}^{7/2}}\right )\,\left (-c\,d^2\,g^2+8\,c\,d\,e\,f\,g-3\,b\,d\,e\,g^2+8\,c\,e^2\,f^2-12\,b\,e^2\,f\,g+15\,a\,e^2\,g^2\right )}{4\,e^{3/2}\,{\left (d\,g-e\,f\right )}^{7/2}}-\frac {\frac {2\,\left (c\,f^2-b\,f\,g+a\,g^2\right )}{d\,g-e\,f}+\frac {{\left (f+g\,x\right )}^2\,\left (-c\,d^2\,g^2+8\,c\,d\,e\,f\,g-3\,b\,d\,e\,g^2+8\,c\,e^2\,f^2-12\,b\,e^2\,f\,g+15\,a\,e^2\,g^2\right )}{4\,{\left (d\,g-e\,f\right )}^3}+\frac {\left (f+g\,x\right )\,\left (c\,d^2\,g^2+8\,c\,d\,e\,f\,g-5\,b\,d\,e\,g^2+16\,c\,e^2\,f^2-20\,b\,e^2\,f\,g+25\,a\,e^2\,g^2\right )}{4\,e\,{\left (d\,g-e\,f\right )}^2}}{e^2\,{\left (f+g\,x\right )}^{5/2}-{\left (f+g\,x\right )}^{3/2}\,\left (2\,e^2\,f-2\,d\,e\,g\right )+\sqrt {f+g\,x}\,\left (d^2\,g^2-2\,d\,e\,f\,g+e^2\,f^2\right )} \]

[In]

int((a + b*x + c*x^2)/((f + g*x)^(3/2)*(d + e*x)^3),x)

[Out]

(atan(((f + g*x)^(1/2)*(e^4*f^3 - d^3*e*g^3 + 3*d^2*e^2*f*g^2 - 3*d*e^3*f^2*g))/(e^(1/2)*(d*g - e*f)^(7/2)))*(
15*a*e^2*g^2 - c*d^2*g^2 + 8*c*e^2*f^2 - 3*b*d*e*g^2 - 12*b*e^2*f*g + 8*c*d*e*f*g))/(4*e^(3/2)*(d*g - e*f)^(7/
2)) - ((2*(a*g^2 + c*f^2 - b*f*g))/(d*g - e*f) + ((f + g*x)^2*(15*a*e^2*g^2 - c*d^2*g^2 + 8*c*e^2*f^2 - 3*b*d*
e*g^2 - 12*b*e^2*f*g + 8*c*d*e*f*g))/(4*(d*g - e*f)^3) + ((f + g*x)*(25*a*e^2*g^2 + c*d^2*g^2 + 16*c*e^2*f^2 -
5*b*d*e*g^2 - 20*b*e^2*f*g + 8*c*d*e*f*g))/(4*e*(d*g - e*f)^2))/(e^2*(f + g*x)^(5/2) - (f + g*x)^(3/2)*(2*e^2
*f - 2*d*e*g) + (f + g*x)^(1/2)*(d^2*g^2 + e^2*f^2 - 2*d*e*f*g))